The n-Category Café

Roald was originally thinking about Ezra Getzler’s approach to operads in Operads revisited, and needed to generalize a result of Getzler’s on when the left Kan extension of a symmetric monoidal functor along a symmetric monoidal functor is itself a symmetric monoidal functor.

If you are of that kind of persuasion, you will be aware that symmetric monoidal categories are the pseudo-algebras for the free symmetric strict monoidal category $2$-monad on ${Cat}$, the $2$-category of categories, and that symmetric monoidal functors are the algebra maps for this $2$-monad.

You might then consider the situation of a $2$-monad $T$ on a $2$-category, and ask when the left Kan extension of a map of $T$-algebras along a map of $T$-algebras is again a $T$-algebra map.

Here are three examples of algebras for a $2$-monad that Roald considers.

1. Ordered compact Hausdorff spaces are algebras for the ultrafilter $2$-monad on $2$-Cat. In this case the question becomes, when is the left Kan extension of a continuous order preserving map along another such map also continuous and order preserving?

2. Double categories can be considered as algebras for a certain $2$-monad on a $2$-category of internal categories in a specific presheaf category. (Yes, I find that a mouthful.)

3. Similarly, monoidal globular categories of Batanin are algebras for some $2$-monad.

Going back to the symmetric strict monoidal category monad, this monad actually extends from the $2$-category $Cat$ to the the proarrow equipment of categories, functors and profunctors. Mike wrote a nice post here at the Café on Equipments and what they have to do with limits.

It turns out that there are several examples of $2$-monads on equipments, with interesting algebras and Roald looks at conditions necessary for when, given such a $2$-monad $T$, the left Kan extension of a map of $T$-algebras along map of $T$-algebras is a map of $T$-algebras.

I should at this point say something precise about the main result: given a ‘right suitable normal’ monad $T$ on a closed equipment $K$, Roald defines ‘right colax $T$-promorphisms’ which, together with the usual colax T-algebra maps, form an equipment $T\text{-rcProm}$. The main theorem is the following.

Theorem: Let $T$ be a ‘right suitable normal’ monad on a closed equipment $K$. The forgetful functor $UT: T\text{-rcProm} \to K$ ‘lifts’ all weighted colimits. Moreover its lift of a weighted colimit $colim_J d: B\to M$, where $d: A \to M$ is a pseudomorphism and $J : A \to B$ is a right pseudopromorphism, is a pseudomorphism whenever the canonical vertical cell $$colim_{T} (m \circ Td) \Rightarrow m\circ T(colim_J d)$$ is invertible, where $m: TM \to M$ is the structure map of $M$.